Theorem rexlimd

Description: Deduction form of rexlimd . For a version based on fewer axioms see rexlimdv . (Contributed by NM, 27-May-1998) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof shortened by Wolf Lammen, 14-Jan-2020)

	Ref	Expression
Hypotheses	rexlimd.1	⊢ Ⅎ 𝑥 𝜑
	rexlimd.2	⊢ Ⅎ 𝑥 𝜒
	rexlimd.3	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
Assertion	rexlimd	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rexlimd.1	⊢ Ⅎ 𝑥 𝜑
2		rexlimd.2	⊢ Ⅎ 𝑥 𝜒
3		rexlimd.3	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
4	2	a1i	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
5	1 4 3	rexlimd2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )